Hotelling's T-squared distribution

In statistics, particularly in hypothesis testing, the Hotelling's T-squared [|distribution], proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution.
The Hotelling's t-squared statistic is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.

Motivation

The distribution arises in multivariate statistics in undertaking tests of the differences between the means of different populations, where tests for univariate problems would make use of a t-test.
The distribution is named for Harold Hotelling, who developed it as a generalization of Student's t-distribution.

Definition

If the vector is Gaussian multivariate-distributed with zero mean and unit covariance matrix and is a random matrix with a Wishart distribution with unit scale matrix and m degrees of freedom, and d and M are independent of each other, then the quadratic form has a Hotelling distribution :
It can be shown that if a random variable X has Hotelling's T-squared distribution,, then:
where is the F-distribution with parameters p and m − p + 1.

Hotelling ''t''-squared statistic

Let be the sample covariance:
where we denote transpose by an apostrophe. It can be shown that is a positive definite matrix and follows a p-variate Wishart distribution with n − 1 degrees of freedom.
The sample covariance matrix of the mean reads.
The Hotelling's t-squared statistic is then defined as:
which is proportional to the Mahalanobis distance between the sample mean and. Because of this, one should expect the statistic to assume low values if, and high values if they are different.
From the distribution,
where is the F-distribution with parameters p and n − p.
In order to calculate a p-value, note that the distribution of equivalently implies that
Then, use the quantity on the left hand side to evaluate the p-value corresponding to the sample, which comes from the F-distribution. A confidence region may also be determined using similar logic.

Motivation

Let denote a p-variate normal distribution with location and known covariance. Let
be n independent identically distributed random variables, which may be represented as column vectors of real numbers. Define
to be the sample mean with covariance. It can be shown that
where is the chi-squared distribution with p degrees of freedom.
Alternatively, one can argue using density functions and characteristic functions, as follows.

Two-sample statistic

If and, with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define
as the sample means, and
as the respective sample covariance matrices. Then
is the unbiased pooled covariance matrix estimate.
Finally, the Hotelling's two-sample t-squared statistic is

Related concepts

It can be related to the F-distribution by
The non-null distribution of this statistic is the noncentral F-distribution
with
where is the difference vector between the population means.
In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation,,
between the variables affects. If we define
and
then
Thus, if the differences in the two rows of the vector are of the same sign, in general, becomes smaller as becomes more positive. If the differences are of opposite sign becomes larger as becomes more positive.
A univariate special case can be found in Welch's t-test.
More robust and powerful tests than the Hotelling's two-sample test have been proposed in the literature, see for example the interpoint distance based tests which can be applied also when the number of variables is comparable with, or even larger than, the number of subjects.